Request timed out when upgrading MacOs to Catalina from Mojave

Read the log, look for the problem, try to fix it. It would take me hours to read through what you posted. Did you really think one of us would do that? Those logs are for the Apple engineers.

Please read Writing an effective Apple Support Communities question. We are here to help. We are all users. We are not mind-readers nor clairvoyants. We only know what you write down and tell us. Be specific and thorough. Include your hardware information and installed version of macOS.

My first thought is the disk has been corrupted as a result of the failed Catalina installation. If reinstalling Mojave is what you want to do, then try this:

Reinstall El Capitan or Later Without Erasing Drive

Please be sure you back up, if possible.

1. Restart the computer. Immediately, at or before the chime, hold down the Command and R keys until the Apple logo and progress bar appear. Wait until the Utility Menu appears.
2. Select Disk Utility and click on the Continue button.
3. Select the indented (usually, Macintosh HD) volume entry from the side list.
4. Click on the First Aid button in Disk Utility's toolbar. Wait until the Done button activates, then click on it.
5. Quit Disk Utility and return to the Utility Menu.
6. Select Install OS X and click on the Continue button.

Once you get Mojave back create a complete backup - one that is bootable. If you use Time Machine be sure you are making a complete backup including all system files. A Time Machine backup is bootable if it's a full backup.

Now, you can try installing Catalina.

If I had to guess, I would guess a bad download. Try re-downloading using Ethernet. Or you can try a network recovery. Again, connect using Ethernet, not Wi-Fi.

Internet/Network Recovery of El Capitan or Later without Erasing the Disk

     If possible back up your files before proceeding.

1. Restart the computer. Immediately after the chime hold down the (Command-Option-R) keys until a globe appears.
2. The Utility Menu will appear in from 5-20 minutes. Be patient.
3. Select Disk Utility and click on the Continue button.
4. When Disk Utility loads select the indented target volume (usually named Macintosh HD) from the side list.
5. Click on the First Aid button in Disk Utility's toolbar.
6. When the check is completed without errors, quit Disk Utility and return to the Utility Menu.
7. Select Install OS X and click on the Continue button.

By not erasing the disk the network recovery should simply reinstall macOS without erasing the disk.

Yes, but what you posted was far to much and came with way too little information about where to look in the log. The BridgeOS is installed for laptops that have the TouchBar, and it's part of macOS. Does your MBP have a TouchBar?

If log entries are needed to determine a problem, then it is best to wait until the person helping you asks to see a specific log entry. Usually, it would be targeted to a date and time that a problem occurred. Also, you need to provide complete information about your hardware and version of macOS installed. Then provide a clear description of your problem as it occurred with any error messages that were displayed. It also help to describe anything you've already tried and what happened when you tried them. The more information you can provide the better we can understand what was going on. Keep in mind that we aren't present to see what you did. You should let whoever is helping you specifically ask for any log entries, crash logs, panic reports, etc. before posting them. Log entries are very long and often hard to understand unless one is an Apple engineer. I am not a programmer or software engineer. Crash logs, for example, are virtually useless without special hardware and software neither you nor us have.

Before you try installing Catalina again please be sure you have a bootable backup of your current working system that you can restore if needed.

That is exactly what I posted just prior. I have no idea why Software Update did not work on your computer. Not enough information with which to work.

I hope you do understand that Command-Option-R and Option-⌘-R are the same? No need for different servers. The installation servers know by the keyboard shortcut you've chosen what to install.